The Unifying Role of Extractors Extractors can be viewed as types of: Hash Functions Expander Graphs Samplers Pseudorandom Generators Error-Correcting Codes Unify the theory of pseudorandomness.

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This Tutorial Is framed around connections between extractors & other objects. Well use these to: Gain intuition for the definition. Describe a few applications. Hint at the constructions. Many omissions. For further reading: N. Nisan and A. Ta-Shma. Extracting randomness: a survey and new constructions. Journal of Computer & System Sciences, 58 (1):148-173, 1999. R. Shaltiel. Recent developments in explicit constructions of extractors. Bulletin of EATCS, 77:67-95, June 2002. S. Vadhan. Course Notes for CS225: Pseudorandomness. http://eecs.harvard.edu/~salil

The [NZ93] Paradigm An approach to constructing extractors: 1.Given a general source X 2.Convert it to a block source (X 1,X 2 ) can use part of the seed for this may want many blocks (X 1,X 2, X 3,...) 3.Apply block extraction (using known extractors, e.g. almost pairwise independence) Still useful today it improves extractors, e.g. [RSW00] How to do Step 2?? get a block by randomly sampling bits from source... harder as min-entropy gets lower.

When does this work? When PRG has a black-box proof: for any function f and any statistical test T, i.e. if PRG construction relativizes [Mil99,KvM99] Almost all PRG constructions are of this form. Partial converse: If E XT is an explicit extractor and f has high description (Kolmogorov) complexity relative to T, then E XT ( f, ) is pseudorandom for T.

Conclusions The many guises of randomness extractors extractors, hash fns, expanders, samplers, pseudorandom generators, error-correcting codes translating ideas between views very powerful! increases impact of work on each object The study of extractors many applications many constructions: information theoretic vs. reconstruction paradigm optimality important

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Some Research Directions Exploit connections further. Optimality up to additive constants. Single, self-contained construction for all ranges of parameters. ( [SU01] comes closest.) Study randomness conductors. When can we have extractors with no seed? important for e.g. cryptography w/imperfect random sources. sources with independence conditions [vN51,Eli72,Blu84,SV84, Vaz85, CG85,CGH+85,BBR85,BL85,LLS87,CDH+00] for efficient sources [TV02]